New Results on Singularly Perturbed Fractional KdV and KdV-Burgers Equations
Abstract
This paper investigates traveling wave solutions for singularly perturbed fractional KdV equation and KdV– Burgers equation with a linear confinement term using Caputo fractional derivatives of order α ∈ (3, 4). Using the tanh method, we first derive exact solutions for the classical cases, recovering the well-known hyperbolic secant soliton for the KdV equation and a tanh-profile solution for the KdV–Burgers equation with confinement term. For the singularly perturbed cases, we employ matched asymptotic expansions to construct solutions that capture both the localized wave core and the algebraic decay in the outer region induced by the fractional derivatives. The inner case retains the classical soliton structure, and the outer solution exhibits a power-law tail of order ξ ^(3−α), reflecting the nonlocal effects of fractional derivatives. Our main results reveal that the fractional perturbation modifies the wave’s decay behavior, giving rise to nonlocal effects absent from the classical formulation.
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